It seems to me that the explanation concerning the refutation from a stricter side is not a sufficient example of disagreeing with all the medieval authorities (Rishonim) boldly and forcefully (and perhaps it is like that raven that brings fire to its nest). There the idea is brilliant, but in my opinion it does not stand up to the test of the sources, and even conceptually there is room to dig into its mountains. I elaborated on this in its place, namely column 346, and I will repeat it here with some added polish. And if here it is out of place and out of time, still it will remain in safekeeping for me, and I will offer it when the time comes.

A. In Sotah 29b, the later generation asks a refutation from a stricter side even though the case being derived also has its own stricter side. That is, one can refute from a stricter side against a common-denominator argument that began from an a fortiori inference, even though in such a case there is no basis for the refuter to think that the shared factual characteristic of the two source cases does not exist in the derived case. [And in Nachmanides on Ketubot 32a I saw that he cites Rashi in Sanhedrin, that one refutes from a stricter side only if the derived case does not also have a stricter side, and that fits your approach. But Nachmanides there raises a similar difficulty against him from Pesachim and resolves it. And this still requires examination in light of the Sotah cited above.] On its face this is a conclusive difficulty.

B. It is very, very far-fetched to say that one who refutes from a stricter side thereby overturns many derivations throughout the Talmud (which come up in a search for “and the law returns,” and I can spell out the contents of the passages if necessary), for example: in Zevachim 15b they derive that a non-priest who received the blood of the sacrifice invalidates it, by a common-denominator argument from a blemished priest and a mourner, and the refutations there are halakhic. In Kiddushin 20b a verse is needed to teach that one who consecrates an inherited field may redeem it in halves, because otherwise there would be a common-denominator argument to the contrary from one who sells an inherited field and one who sells a house in a walled city [and more], and the refutations there are halakhic.
And if someone were to refute from a stricter side in every single place where the refutations are halakhic, from where would he derive all these laws? Or what would he do with the verses that come to exclude that derivation? And how is it that the Gemara never takes his view into account anywhere? Clearly, then, this is not a fixed method, but rather each derivation requires separate judgment. And to my mind this is strong evidence, because what is gained by an explanation that takes four appearances of “a stricter side” in the Talmud and explains that in all of them the refutations are halakhic, while ignoring other appearances of halakhic refutations (presumably more numerous) in which “a stricter side” was not invoked?
And another quasi-proof: in Menachot 60b Rabbi Yehuda has a common-denominator derivation with halakhic refutations, and Rabbi Shimon does not accept the derivation, and the Gemara discusses what they disagree about and attributes the dispute to something fairly questionable. Why didn’t it simply say: Rabbi Yehuda holds that one does not refute from a stricter side, as it says in the passage in Makkot 4 that you dissected into pieces, while Rabbi Shimon holds that one does refute from a stricter side?

C. And I bring further proof that this is not a fixed method, from the words of the amora Ulla. In Ketubot 32a they refute from a stricter side according to Ulla, whereas in Bava Kamma 88a Ulla himself derives a common-denominator argument with halakhic refutations and does not refute from a stricter side [and also in Makkot 4b Ulla, according to Rabbi Yehuda, does not refute from a stricter side]. Thus we learn that each derivation depends on its own context, and no one maintains a fixed method either to refute from a stricter side everywhere or never to do so anywhere.

D. And if you wish, say it on conceptual grounds. Even according to your approach, in practice one does not refute from a stricter side. That is, the refuter, even though he stands in the role of defense, cannot invent that there is some shared factual characteristic behind two different stringencies and claim that it does not exist in the derived case. Rather, he must proceed as in an archetypal derivation and assume that if there is a shared factual characteristic in the source cases, then it exists in the derived case as well, and therefore the derived case should also receive the new law. And this explanation is so intuitive that it is hard to think that one who refutes from a stricter side would disagree with it.